Heat transfer and Heat Equations

3-3. Use the methods of vectorcalculus to derive the general heat conduction equation (Hint: Apply the first law to a volume V with surface S. and use the Gauss divergence theorem to Convert the surface integral of heat flow across S to a
volume integral over V.)
The cylindrical and spherical coordinate systems are examples of orthogonal curilinear coordinates. In general. we can denote these coordinates by ..... which are defined by specifying the Cartesian coordinates. .... as
.....
A coordinate system is orthogonal when the three families of surfaces ii Const. u, -= Consi, u3 = Const are orthogonal to one another. The figure shows an elemental paraliepiped whose faces coincide with planes u1 or or ui = Const. with edge lengths h1du h2dx,2. h1dii1 where h1. h, h are called the metric coefficseiirs. The length of a diagonal is given by
ds2 = hdu + hdu + h,du
in terms of these coordinates, the components of the temperature gradient are....

(i) Identify the metric coefficients for the cylindrical coordinate system, and hence write down V2T in cylindrical coordinates.
(ii) Repeat for the spherical coordinate system.

textbook: A F Mills, Heat Transfer, Prentice Hall, 1999

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